×

Orthogonal series representations for generalized functions. (English) Zbl 0647.46037

The paper contains a theory of expansion of certain Schwartz distributions with respect to series of orthonormal families of functions in weighted \(L^ p\) spaces, i.e. \(A_{L_ wp}\) spaces. Conditions for convergence of orthogonal series in these spaces have been found. Also their dual spaces have been studied and characterized. The results include theorems on expansions into series of orthogonal polynomials, such as Hermite, Laguerre, Legendre etc. polynomials. The theory has applications, for instance, in solving some differential equations in distribution spaces.
Reviewer: P.Kruszyński

MSC:

46F10 Operations with distributions and generalized functions
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ditzan, Z, Summability of Hermite polynomial expansions of generalized functions, (), 129-139 · Zbl 0194.08805
[2] Dube, L.S, On finite Hankel transformation of generalized functions, Pacific J. math., 62, 365-378, (1967) · Zbl 0329.46044
[3] ()
[4] Friedman, A, Generalized functions and partial differential equations, (1963), Prentice-Hall N.J · Zbl 0116.07002
[5] Judge, D, On Zemanian’s distributional eigenfunction transforms, J. math. anal. appl., 34, 187-201, (1971) · Zbl 0224.46047
[6] Pandey, J.N; Pathak, R.S, Eigenfunction expansion of generalized functions, Nagoya math. J., 72, 1-25, (1978) · Zbl 0362.34018
[7] Pathak, R.S, Eigenfunction expansion of generalized functions, C. R. math. report acad. sci., Canada, 5, 6, 281-286, (1983) · Zbl 0527.47014
[8] Pathak, R.S, Summability of Laguerre polynomial expansion of generalized functions, J. inst. math. appl., 21, 171-180, (1978) · Zbl 0379.40013
[9] Pathak, R.S; Singh, O.P, Finite Hankel transforms of distributions, Pacific J. math., 99, 439-458, (1982) · Zbl 0484.46039
[10] Polard, H, The Mean convergence of orthogonal series of polynomials, (), 8-10
[11] Polard, H, The Mean convergence of orthogonal series I, Trans. amer. math. soc., 62, 387-403, (1947)
[12] Polard, H, The Mean convergence of orthogonal series II, Trans. amer. math. soc., 63, 355-367, (1948)
[13] Schwartz, L, Théorie des distributions, (1978), Hermann Paris
[14] Sneddon, I.N, The use of integral transforms, (1972), McGraw-Hill New York · Zbl 0265.73085
[15] Sobolev, S.L, Applications of functional analysis in mathematical physics, (1963), Amer. Math. Soc Providence, RI · Zbl 0123.09003
[16] Wing, G.M, The Mean convergence of orthogonal series, Amer. J. math., 72, 792-807, (1950) · Zbl 0045.33801
[17] Zemanian, A.H, Orthonormal series expansions of certain distributions and distributional transform calculus, J. math. anal. appl., 14, 263-275, (1966) · Zbl 0138.37804
[18] Zemanian, A.H, Generalized integral transformations, (1968), Interscience New York · Zbl 0181.12701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.